# Rep of Non-Commutative Monoids

Let M be a non-commutative monoid. It is possible that all representation of M are one dimensional ??

(for groups the answer is negative. Take a non zero x=[a,b]. Take a representation where x does not act trivially. This rep can not be one dimensional. But on monoids I can not take the commutator)

• Do you mean irreducible? Otherwise all monoids have reps of all dimensions. Mar 28 '14 at 17:36

The question is vague. I am assuming you want finite monoids over the complex field, although I could answer over any field, and I am assuming you want irreducible reps. There are noncommutative monoids whose irreducible representations are all 1-dim. I characterized with Almeida, Margolis and Volkov all such monoids (http://arxiv.org/pdf/math/0702400.pdf). The simplest example is take the three element monoid consisting of $1,a,b$ where 1 is the identity and $xy=x$ if $x$ is not 1. You can also take the path semigroup of an acyclic quiver.
The theorem is that a finite monoid $M$ has only 1-dimensional irreps iff
1. The group of units of $eMe$ is abelian for all idempotents $e$.
2. If $e,f$ are idempotents and $MeM=MfM$ then $ef$ is idempotent and $MefM=MeM$.